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Let $G$ be discrete Abelian group and denote by $\widehat G$ the Pontryagin-Van Kampen dual of $G$. I was reading in a paper due to Justin Peters that Fourier Transform induces a bijection between the following sets of functions:

(1) $L^1(G)^+\cap \mathcal P(G)$ (continuous, non-negative, positive-definite and absolutely integrable functions $G\to \mathbb C$);

(2) $L^1(\widehat G)^+\cap \mathcal P(\widehat G)$ (continuous, non-negative, positive-definite and absolutely integrable functions $\widehat G\to \mathbb C$).

Peters says that this is easy to deduce from Fourier inversion theorem but it does not seem so elementary to me. Can anyone help me? Is this true for any LCA group?

Is there any analog if we start with a compact non-commutative group and we use the Tannaka-Krein duality?

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